An infinite line charge of uniform electric charge density `lambda` lies along the axis of an electrically conducting infinite cylindrical shell of radius R. At time `t=0`, the space inside the cylinder is filled with a material of permittivity `epsilon` and electrical conductivity `sigma`. The electrical conduction in the material follows Ohm's law. Which one of the following graphs best describes the subsequent variation of the magnitude of current density j(t) at any point in the material ?

Options

A.
B.
C.
D.

Answer

C

Explanation

(c) `J = I/(2pirl) = (dV//dR)/(2pirl)` ……(i)
`dR = rho(dr)/(2pirl) = 1/sigma xx (dr)/(2pirl)`…………(ii)
Now, `E = -(dV)/(dr)`
`:. dV = -Edr = -(lambda/(2piepsilon r)) dr` …….(iii)
From (i), (ii) and (iii)
`J = 1/(2pirl) [ (lambdadr)/(2lambdaepsilonr) xx (sigma2pirl)/(dr)] = (lambdasigma)/(2piepsilonr)` .........(iv)
Also `I = (dV)/(dR) = (-lambda)/(2piepsilonr) dr xx (sigma xx 2pirl)/(dr) = (-lambdasigmal)/epsilon`........(v)
Here negative sign signifies that the current is
decreasing
But ` I = (d(q))/(d) = (d(lambda l))/(dt) = l(dlambda)/(dt)` ....... (vi)
From (v) and (vi)
`l (dlambda)/(dt) = -(lambdasigmal)/(epsilon) rArr (dlambda)/(lambda) = (-sigma)/(epsilon l) dt`
On integrating
`int_(lambda_0)^lambda (dlambda)/lambda = -sigma/epsilon int_(0)^t dt`
`:. log_e lambda/lambda_0 = -(sigmat)/epsilon :. lambda = lambda_0 e^(-sigma/epsilon t)`
Substituting this value in (iv) we get
`J = (sigmalambda_0)/(2piepsilonr) e^(-sigma/epsilon t)`.